I have done this too, although it was not 'officially' a battle bot, but it had lot's of elements in it.
I suggest to use rubber 'tank' tires (i don't know how to call them), you know, those bands that are on war-tanks:
http://farm1.static.flickr.com/24/35276932_417dab14ac_m.jpg.
For the wheels, you don't have to use as much as in the picture. Four in total will do (two in the back, two in the front), just like a car. But of course the wheels drive the rubber band, and the wheels get driven by an electromotor.
For the machinery (barrel to shoot things with, ejection-cillinder to use as a 'punching'-gun, or whatever), you can use pneumatics. It is very simple, but make sure the mechanics (working with steel, cutting, bending, etc.) are DEAD ACCURATE in order to make the whole work fluently. The drive (electrical, pneumatical) isn't very hard.
Good luck